Factorization of density
correlation functions for
clusters touching the sides of
a rectangle
Peter Kleban
University of Maine
Jacob J. H. Simmons
University of Chicago
Robert M. Ziff
University of Michigan
Support: NSF DMR–0536927 (PK), EPSRC Grant EP/
D070643/1 (JJHS), and DMS–0553487 (RMZ)
• Introduction and background
• Clusters touching the sides of a rectangle
• Choice of co–ordinates
• Solving the PDEs
• Almost exact factorizations
A Guide for the (conformally)
Perplexed:
Consider percolation in the upper half plane, in the
continuum (field theory) limit. The probability that the
interval [u1, u2] on the real axis is connected to a small
circle of size ε around the point w = u + i v is then
ε
↓
w
↑
u1
u2
This is given by
P(u1 , u2 , w) =
2
5/48
c1,2 c1/2,0 !
f (u1 , u2 , w)
Here the (dimensionless) constants cm,n depend on the
percolation model (lattice symmetry, type of percolation),
and are thus “nonuniversal”. There is such a constant for
each identified point (the subscripts will be explained below).
We consider ratios in which the cm,n and ε factors cancel out,
so the ratios are “universal” (independent of the particular
percolation model) and finite as ε → 0.
Conformal field theory (CFT) calculates the function
f(u1, u2, w). This is a “correlation function”. In the case just
mentioned
we showed previously that
f(u1, u2, w) = v–5/48 sin1/3(ζ/2),
where the angle ζ is as illustrated:
w=u+iv
u1
u2
In CFT notation, the present correlation function (cf) is
written
f(u1, u2, w) = <ϕ1,2(u1) ϕ1,2(u2) ϕ1/2,0(w)>H
where the H denotes the upper half-plane. Here the ϕm,n are
conformal “operators”. One usually computes such a cf in the
full plane by introducing an “image” operator at the reflected
point w̄ = u - i v, following Cardy, ie
f(u1, u2, w) = <ϕ1,2(u1) ϕ1,2(u2) ϕ1/2,0(w) ϕ1/2,0(w̄)>
When the ϕm,n have integer indices, the cfs that they appear
in satisfy DEs. That allows us to determine the cfs.
The conformal “operators” of use here are
1.) φ(1,2)(u), which implements a change from fixed
(“wired”) to free boundaries at u.
u
2.) φ(3/2,3/2)(w) = φ(1/2,0)(w), the “magnetization” operator.
This measures the density of clusters at the point w.
Conformal dimensions (@ c=0): h(1,2) = 0, h(3/2,3/2) = 5/96.
Clusters touching the sides of a
rectangle:
We consider percolation in a rectangle with the vertical edges
“wired” (fixed bcs there). Take z = x + i y as the co–
ordinate. Then let PL(z) (resp. PR(z), PLR(z)) be the
probability of a cluster touching the point z and the left (resp.
right, left & right) sides of the rectangle. Some examples:
z
PL(z)
z
z
PL(z), PR(z), PLR(z)
Then consider the ratio
PLR (z)
C(z) = !
PL (z)PR (z)Πh
Here πh is Cardy’s (horizontal) crossing probability. The
ratio C(z) is independent of the cm,n and ε factors mentioned,
and therefore is universal and can be calculated from CFT.
The calculation is not so simple, since six-point cfs must be
determined–four points for the corners of the rectangle, and
two for the operator at z and its image.
Why C(z)?
a. Previous results on a related ratio (with points rather
than intervals) which factorizes exactly (i.e. C is
independent of z). πh “improves” things, removing most
aspect ratio dependence (it also make the ratio more
homogeneous).
b. Bob Ziff’s numerical results. He found that C(z) is
1. constant to within a few % everywhere in the rectangle–
i.e. PLR(z) “almost” factorizes
2. a function of x only (i.e. it is independent of the vertical
coordinate).
Our calculation verifies 1. and shows that 2. holds exactly.
The cf that must be determined is
<φ(1,2)(u1) φ(1,2)(u2) φ(1,2)(u3) φ(1,2)(u4)φ(1/2,0)(w)φ(1/2,0)(w̄) >
w = u + i v is the half–plane co–ordinate.
This is a complicated function. Aside from an algebraic prefactor, it depends on a function F of three independent
cross-ratios.
We write equations for arbitrary central charge
(equivalently, arbitraryκ) but for brevity present
conclusions for percolation only (c = 0,κ = 6)
Because of the φ(1,2) operators, the cf is annihilated
by certain second–order operators. For ex., the
operator associated with u1 is
Because of this, the factor F satisfies second–order
PDEs. We can choose the cross-ratios so that
Letting {u1, u2, u3, u4}→{0, λ, 1, ∞}, gives
F → F(w,w̄, λ). Note that λ determines the aspect
ratio r of the rectangle.
Proper choice of co-ordinates greatly simplifies the
PDEs, as the numerical result suggests.
Choice of co–ordinates:
We next transform from rectangle coordinates z = x + i y to
the upper half–plane via a Schwarz–Christoffel map
w(z) = m sn(z|K’(m))2
Here
1. K’(m) = K(1-m), with K the complete elliptic integral of
the first kind
2. sn( |m) is the Jacobi elliptic function. The elliptic
parameter m is defined by r = K(m)/K’(m), r being the aspect
ratio of the rectangle.
z
x
ξ
The next step is key. We chose real coordinates which
reflect the rectangular symmetry, namely
ξ = sn(x|K’(m))2
ψ = sn(y|K’(1-m))2.
ξ is the half–plane image of x. ψ is more complicated.
Exchanging x↔y (i.e. z↔iz̄ ), and rescaling the rectangle to
preserve its aspect ratio r, makes ψ the half–plane image of y.
The transformation to the upper–plane then becomes
In these co–ordinates, the PDE (with λ = m) is
(thank you, Mathematica!)
Solving the PDEs, origin of
the y–independence, and
identifying cluster
configurations:
How do we handle this?
1.) Using the symmetries of the rectangle: mirror about x =
r/2, mirror about y = 1/2, and rotation by 900 (with change
of aspect ratio r → 1/r) gives three additional PDEs.
Because of the symmetry of our choice of ξ and ψ, these are
the same equations that arise from applying CFT at the
other φ(1,2) operators.
These symmetries translate, respectively, into
2.) There is a linear combination of the four equations which
gives
(2. is the origin of the y–independence of C(z) discussed
above.)
3.) Thus all solutions must be of the form
(A solution of the form g(m) can’t satisfy the original PDEs.)
4.) Further, the symmetry{ψ↔ξ, m↔1-m} implies
F(ξ, ψ,m) = G(ξ,m) + G(ψ,1-m).
5.) Redefining G as
G(ξ,m) =
substituting into the DEs, and taking linear combinations
that allow us to cancel out all factors of ψ we arrive at
where s = m and t = m ξ, the standard form of Appell’s
hypergeometric DEs for the Appell function F1. (Note we
write this for arbitraryκ.)
The Appell function F1 is a two–variable generalization of the
hypergeometric function:
6.) There is a three–dimensional solution space, including
the five convergent Frobenius series
7.) Making use of this, we find five solutions for the original
six–point correlation function. That is still not quite the
whole story: since each of these is a Frobenius series, it
corresponds to a conformal block (function associated with a
single term in the CFT operator product expansion). But
what we really want are the functions associated with each
cluster configuration of interest:
LR, L, R
R
L
8.) Some analysis involving vertex operators (translation:
integral representations) and various limits gives for our five
solutions (here Π is the weight of the indicated
configuration)
GI
GII
GIII
GIV
GV
∼
∼
∼
∼
∼
ΠR
ΠLR + ΠR
ΠLR + ΠL + ΠR
ΠLR + ΠL
ΠL
9.) Finally, transforming to a rectangle (and noting that
corner operators enter) we find
ΠLR
ΠR
ΠL
= f (x, y) (GII (ξ) − GI (ξ))
= f (x, y) GI (ξ)
= f (x, y) GV (ξ)
with the common factor
f(x,y) =
(As above, K´=K(1-m), with K the complete elliptic
integral.)
The common factor f(x,y) cancels out of the ratio, so that
C(z) only depends on m (parameterizing the aspect ratio) and
ξ, which is independent of y. Thus for a given rectangle
C = C(x),
demonstrating that the original numerical observation is
exact.
Furthermore, the whole derivation can be generalized to
arbitrary central charge (equivalently, arbitrary n orκ). The
(approximate) factorization generalizes as well. In doing
this, the cfs that enter C must generalized as well.
Note added:
Dmitri Beliaev and Konstantin Izyurov have recently
obtained a rigorous derivation of these PDEs for the case
SLE6 with one interval of infinitesimal length.
Conclusion:
Results from conformal field theory show that a certain
ratio of correlation functions in 2-D critical rectangles
almost factorizes exactly, with the deviation from exact
factorization depending on only one co–ordinate. This
follows from a solution of the CFT PDEs in properly
chosen co–ordinates.
YAPPS
YAPPS
Yet
Another
Peculiar
Percolation
Symmetry